A functor-valued invariant of tangles

Mikhail Khovanov

Type: Article

Publication Date: 2002-09-06

Citations: 281

DOI: https://doi.org/10.2140/agt.2002.2.665

Abstract

We construct a family of rings.To a plane diagram of a tangle we associate a complex of bimodules over these rings.Chain homotopy equivalence class of this complex is an invariant of the tangle.On the level of Grothendieck groups this invariant descends to the Kauffman bracket of the tangle.When the tangle is a link, the invariant specializes to the bigraded cohomology theory introduced in our earlier work.

Locations

Algebraic & Geometric Topology - View - PDF
arXiv (Cornell University) - View - PDF
Project Euclid (Cornell University) - View - PDF
DataCite API - View

Similar Works

Action	Title	Year	Authors
+ PDF Chat	An invariant of tangle cobordisms	2005	Mikhail Khovanov
+	Functors Extending the Kauffman Bracket	2006	John Armstrong
+	FUNCTORS EXTENDING THE KAUFFMAN BRACKET	2009	John Armstrong
+ PDF Chat	Tangle Floer homology and cobordisms between tangles	2020	Akram Alishahi Eaman Eftekhary
+	Link invariants from finite categorical groups and braided crossed modules	2013	João Faría Martins Roger Picken
+	Link invariants from finite categorical groups and braided crossed modules	2013	João Faría Martins Roger Picken
+	Link invariants from finite categorical groups	2016	João Faría Martins Roger Picken
+ PDF Chat	Link invariants from finite categorical groups	2015	João Faría Martins Roger Picken
+	Link invariants from finite categorical groups	2016	João Faría Martins Roger Picken
+	A categorification of the skein module of tangles	2004	Marta Asaeda Józef H. Przytycki Adam S. Sikora
+ PDF Chat	On the Tambara Affine Line	2024	Dave Chan David Mehrle J. D. Quigley Ben Spitz Danika Van Niel
+ PDF Chat	Alexander Representation of Tangles	2015	Stephen Bigelow Alessia Cattabriga Vincent Florens
+ PDF Chat	Kauffman states and Heegaard diagrams for tangles	2019	Claudius Zibrowius
+	Framed tangles and a theorem of Deligne on braided deformations of Tannakian categories	1992	David N. Yetter
+ PDF Chat	Categorification of the skein module of tangles	2006	Marta Asaeda Józef H. Przytycki Adam S. Sikora
+ PDF Chat	Quantum <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" altimg="si1.gif" overflow="scroll"><mml:msub><mml:mrow><mml:mi mathvariant="fraktur">gl</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn><mml:mo stretchy="false">\|</mml:mo><mml:mn>1</mml:mn></mml:mrow></mml:msub></mml:math> and tangle Floer homology	2019	Alexander P. Ellis Ina Petkova Vera Vértesi
+	Alexander representation of tangles	2012	Stephen Bigelow Alessia Cattabriga Vincent Florens
+	Alexander representation of tangles	2012	Stephen Bigelow Alessia Cattabriga Vincent Florens
+	Quillen cohomology of enriched operads	2025	Hoang Le Truong
+	Meta-Monoids, Meta-Bicrossed Products, and the Alexander Polynomial	2013	Dror Bar-Natan Sam Alexandre Selmani

Works That Cite This (280)

Action	Title	Year	Authors
+	Khovanov homology from Floer cohomology	2018	Mohammed Abouzaid Ivan Smith
+	Sutured annular Khovanov-Rozansky homology	2017	Hoel Queffélec David E. V. Rose
+ PDF Chat	The Fukaya category of the pillowcase, traceless character varieties, and Khovanov cohomology	2020	Matthew Hedden Christopher M. Herald Matthew Hogancamp Paul Kirk
+ PDF Chat	The correspondence induced on the pillowcase by the earring tangle	2022	Guillem Cazassus Christopher M. Herald Paul Kirk Artem Kotelskiy
+ PDF Chat	Skein Modules from Skew Howe Duality and Affine Extensions	2015	Hoel Queffélec
+	Skein relations for tangle Floer homology	2016	Ina Petkova C.‐M. Michael Wong
+	An unoriented skein relation for tangle Floer homology	2016	Ina Petkova C.‐M. Michael Wong
+ PDF Chat	A categorification of quantum $\mathrm{sl}(n)$	2010	Mikhail Khovanov Aaron D. Lauda
+ PDF Chat	𝔤𝔩n-webs, categorification and Khovanov–Rozansky homologies	2020	Daniel Tubbenhauer
+	Khovanov homology and categorification of skein modules	2018	Hoel Queffélec Paul Wedrich

Works Cited by This (51)

Action	Title	Year	Authors
+	Triangulated categories in the modular representation theory of finite groups	1998	Jeremy Rickard
+	New Examples of Frobenius Extensions	1999	Lars Kadison
+ PDF Chat	On the decomposition numbers of the Hecke algebra of $G(m, 1, n)$	1996	Susumu Ariki
+	Lectures on Tensor Categories and Modular Functors	2000	Bojko Bakalov Alexander Kirillov
+	Representations of Algebraic Groups	2007	Jens Carsten Jantzen
+	Modular Lie Algebras and their Representations	2020	Helmut Strade
+ PDF Chat	Algebraic structures behind Hilbert schemes and wreath products	2002	Weiqiang Wang
+	Instantons on ALE spaces, quiver varieties, and Kac-Moody algebras	1994	Hiraku Nakajima
+	Triangulated Categories in the Representation of Finite Dimensional Algebras	1988	Dieter Happel
+	Cohomology of sheaves	1984	Birger Iversen